Not sure what you mean by "bypassing public folders" exactly, but you can remove them from the library so they won't show in the homegroup.
You will probably have to use the network option to back up your netbook, if the Starter Edition allows it, and maybe have to share the external drive with the homegroup. Or use normal network access.
Backing up two systems might be a problem, since Windows backup, at least the image, uses the same name for the image backup folders and overwrites a prior one.